This clinical trial studies intra-osseous donor umbilical cord blood and mesenchymal stromal cell co-transplant in treating patients with hematologic malignancies. Giving low doses of chemotherapy and total-body irradiation before a co-transplant of donor umbilical cord blood and mesenchymal stromal cells into the bone (intra-osseous) helps stop the growth of cancer cells. It may also stop the patient's immune system from rejecting the donor's stem cells. The donated stem cells may replace the patient's immune cells and help destroy any remaining cancer cells (graft-versus-tumor effect). Sometimes the transplanted cells from a donor can also make an immune response against the body's normal cells. Giving cyclosporine and mycophenolate mofetil at the time of transplant may stop this from happening.